Question

If a line makes an angle ${\theta }_1,{\theta }_2,{\theta }_3$ which the axis respectively, then $cos2{\theta }_1+cos2{\theta }_2+cos2{\theta }_3=?$

• A. $-4$
• B. $2$
• C. $3$
• D. $-1$

Answer 1

The correct option is B $2$

Let $P(x,y,z)$ be any point on the line OP, D-Ratio of OP are x,y,z. D.R. of normal plane$i.e.,z=0$are$0,0,1$

$\sin {\theta }_1=\frac{z}{\sqrt{x^2+y^2+z^2}}$

similarily for $sin{\theta }_2$ and $sin{\theta }_3$

$\therefore si{n}^2{\theta }_1+si{n}^2{\theta }_2+si{n}^2{\theta }_3$

$=\frac{x^2+y^2+z^2}{x^2+y^2+z^2}=1$

$⟹1-co{s}^2{\theta }_1+1-co{s}^2{\theta }_2+1-co{s}^2{\theta }_3=1$

$\therefore co{s}_1^{\theta }+co{s}_2^{\theta }+co{s}_3^{\theta }=3-1=2$